APPENDIX 



THEORY AND PLOTTING DATA FOR REFRACTION SCALES 



The theory involved in the construction ot the 

 scales shown in figure 5 for plotting refraction 

 diagrams by the wave-advance method is briefly 

 as follows: 



It is desired to plot values of wave advance as a 

 function of the ratio, djLo. By proper spacing 

 of values of djLo, the upper plotting edge of the 

 scale can be made a straight line, hence it can be 

 constructed with considerable accuracy. 



Referring to the following sketch, x represents 

 the base length of the scale and the ordinate at the 

 right-hand side represents the wave advance in 

 deep water; that is, the advance is some multiple 

 n of the deep water wave length, Lo. 



C' 



(4) 



nL„ 



For any particular value of the depth-length 

 ratio, such as dn/Lo, the distance from the left- 

 hand end of the scale to the point where the wave 

 advance is nL„ is, by similar triangles, given by 

 the relationship, 



X nL, 

 or 



■\r -17- ^n 



(1) 



For any chosen length of scale (8 inches for 

 scales A and B, fig. 5) values of X„ for various 

 assumed values of djLo are calculated by the 

 following procedure: 



For shallow water, the wave velocity is 



and 



0^=^^tanh^ 



ZTT Li 



(2) 

 (3) 



'T^Lo/5.l2 

 hence, a combination of equations (2) and (4) gives 



^=tan/i(?^) (5) 



Table 4 gives the steps in computing values of 

 Xn by use of equation (1) and (5). Column (1) 

 shows various values of d/Lo. From Breakers and 

 Surf, H. 0. No. 234, values of L/Lo have been 

 obtained for the corresponding values of d/Lg and 

 tabulated in column (2). Column (3) shows 

 values of d/L which were obtained by dividing 

 column (1) by column (2). Column (4) is the 



Table 4 

 Computations and plotting data for wave refraction scales 



or equation (3) may be written 



44 



